scholarly journals k-TYPE SLANT HELICES FOR SYMPLECTIC CURVE IN 4-DIMENSIONAL SYMPLECTIC SPACE

2020 ◽  
pp. 641
Author(s):  
Esra Cicek Cetin ◽  
Mehmet Bektaş
Keyword(s):  
Differential Equations ◽  
Symplectic Space ◽  
Slant Helix ◽  
Symplectic Curve

In this study, we have expressed the notion of $k$-type slant helix in $4$-symplectic space. Also, we have generated some differential equations for $k$-type slant helix of symplectic regular curves. 

Hamiltonian Duality Equation on Three-Dimensional Problems of Magnetoelectroelastic Solids

2012 ◽  
Vol 268-270 ◽  
pp. 1099-1104
Author(s):  
Xiao Chuan Li ◽  
Jin Shuang Zhang
Keyword(s):  
Variational Principle ◽  
Differential Equations ◽  
Separation Of Variables ◽  
Electric Displacement ◽  
Hamiltonian Formulation ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Coefficient Matrix ◽  
Symplectic Space ◽  
Magnetoelectroelastic Solids

Hamiltonian system used in dynamics is introduced to formulate the three-dimensional problems of the transversely isotropic magnetoelectroelastic solids. The Hamiltonian dual equations in magnetoelectroelastic solids are developed directly from the modified Hellinger-Reissner variational principle derived from generalized Hellinger-Ressner variational principle with two classes of variables. These variables not only include such origin variables as displaces, electric potential and magnetic potential, but also include such their dual variables as lengthways stress, electric displacement and magnetic induction in the symplectic space. Similar to the Hamiltonian formulation in classic dynamics, the z coordinate is treated analogous to the time coordinate so that the method of separation of variables can be used. The governing equations are a set of first order differential equations in z, and the coefficient matrix of the differential equations is Hamiltonian in (x, y).

Characterizations of Spacelike Slant Helices In Minkowski 3-Space

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
İ. Gök ◽  
S. Kaya Nurkan ◽  
K. Ilarslan ◽  
L. Kula ◽  
M. Altinok
Keyword(s):  
Differential Equations ◽  
Subject Classification ◽  
Normal Vector ◽  
Spacelike Curve ◽  
Frenet Equations ◽  
Slant Helix ◽  
Spherical Indicatrix ◽  
Tangent Indicatrix ◽  
Principal Normal Indicatrix ◽  
Binormal Indicatrix

Abstract In this paper, we investigate tangent indicatrix, principal normal indicatrix and binormal indicatrix of a spacelike curve with spacelike, timelike and null principal normal vector in Minkowski 3-space E3 1 and we construct their Frenet equations and curvature functions. Moreover, we obtain some differential equations which characterize for a spacelike curve to be a slant helix by using the Frenet apparatus of spherical indicatrix of the curve. Also related examples and their illustrations are given. Mathematics Subject Classification 2010: 53A04, 53C50.

scholarly journals The Characterization of Affine Symplectic Curves in ℝ4

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 110
Author(s):  
Esra Çiçek Çetin ◽  
Mehmet Bektaş
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Fourth Order ◽  
Symplectic Geometry ◽  
Classical Mechanics ◽  
Symplectic Space ◽  
Frenet Frame ◽  
The Third ◽  
Natural Geometry

Symplectic geometry arises as the natural geometry of phase-space in the equations of classical mechanics. In this study, we obtain new characterizations of regular symplectic curves with respect to the Frenet frame in four-dimensional symplectic space. We also give the characterizations of the symplectic circular helices as the third- and fourth-order differential equations involving the symplectic curvatures.

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